Crystals and Interference
How to use the program
The applet generates the interference figures of non-isotropic media as seen through a
polarization microscope. The following suggestions may give you a starting point..
- If you press START, using the parameters set by the
program, you will see the interference image produced by a uniaxial non-optically active
crystal whose optical axis lies perpendicular to the screen. You can observe two different
kinds of black regions in the figure, isogyres and isochromes. The large triangular
areas forming the shape of a windmill sail result from the setting of the
linear polarizer and analyzer. They are called isogyres (lines of the same inclination of the D-field's polarization plane).
Since xi, the angle between the
polarizer and the analyzer, is set to 0.5 pi, the black regions where the light is not transmitted by the analyzer
correspond to directions along which the light's polarization plane is not altered
by the crystal. This is the case, if the polarization of the
incident ray of light coincides with the polarization plane defined by the crystal symmetry, thus
the ray is not split up into two rays with paths of different refraction indices but keeps its polarization
throughout its whole passage through the crystal. The intersection of the isogyres marks the direction of the
optical axis. The second kind of black regions, the isochromes, are caused by the interference of the two
beams passing through the crystal. With the setting of the default parameters they are
just circles around the center indicating the ray which passes exactly along the optical
- You can check the remarks above by changing the
relative analyzer angle xi to 0.0 pi. Here the exact inverse (photographic negative)
of the original image should appear. What happens to the isogyres if you change the
polarizer angle eta?
- To make the shape of the isogyres more clear choose
the very last setup "draw D-field vector" via the choice menu and press
START. This will generate the vector field of D. Think now about the polarization of the incident ray and
the setting of the analyzer. Along which paths will the light beam be extinguished?
- You can investigate the isochromes
without having to be disturbed by the
pattern of the isogyres by simply switching the setup to "circular
polarizer, circular analyzer".
- Return to the setup "linear polarizer, linear analyzer".
The image produced by a biaxial crystal is generated by
changing the refraction index, making ny different from nx. For realistic images the
change should be in the order 0.001 - 0.01. Start by giving ny a value between nx and nz..
- The angles theta and phi define the orientation of the
crystal´s optical axis with respect to the screen coordinates. Change theta to see what a
rotation of the crystal does to the interference image. At theta= 0.5 pi hyperbolic lines
appear, indicating that the crystals optical axis is now lying parallel to the plane of
the screen. What happens if you interchange the values of nx, ny and nz?
- The type of curves (isochromes) in the
image produced by a biaxial crystal (nx != ny, both < nz) at the angle theta=0 are called lemniscates. The two
center points of a lemniscate indicate the two directions of the two optical axes of the
crystal. The angle between these axes can be calculated to be beta =
ARCTAN(nz/ny*SQRT((ey - ex)/(ez - ey))) where ex,ey,ez are the dielectricities given by
the squares of the refraction indices, that is ex=SQR(nx), ey=SQR(ny), ez=SQR(nz). You can
check this formula, by calculating beta for your values of nx,ny,nz and then rotating the
crystal in such a way that once the first center and then the second center of the
lemniscate lies in the middle of the applets frame. How big was the angle by which you had to rotate?
- To study optical
activity you have to give one of the gyroscopic tensor elements gij a value different from zero.
Optical activity is a phenomenon much smaller than birefringence, so it is best observed in a region, where
the refraction indices coincide. This is the case along the optical axes. If we set nx=ny, the optical axis will be
the z-axis, so the element of the gyration tensor to consider will be gzz.
The most impressive phenomenon due to
optical activity are the Airy spirals. Choose the setup "linear polarizer, circular
analyzer" with nx=ny=1.544, nz=1.553 and theta=0.0, thickness=3.0.
Now set gzz=0.001 and press
START. What happens if you set ny different from nx?
- Note that the
optical activity does not influence at all the graph of the D-vector field, since the polarization planes follow
directly from Fresnels (not extended) equations. The eigen polarizations of an optically active crystal are elliptical,
so what is drawn in the vector field are now not the polarization planes but the principal axes of the elliptical eigen polarizations.
You may think, that since an optically active medium does rotate the
polarization of a traversing beam, it also should affect the crystal's polarization planes. This is not the case, since the
rotation of the polarization is again an effect of splitting and recombining a beam at entrance and exit of the crystal.
- All points listed
above can of course be also checked in the colored version. Here the name isochromes
becomes more evident, they are the curves formed by light of the same wavelength (chroma gr.=color).
- The applet does not have any zoom function, but you can
inspect interesting regions more closely by reducing the crystals thickness
and rotating the crystal to the desired area.
Go on to
Some technical details
Java virtual polarizing microscope, interference figures